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Forces in 2D - Vector Notation (AQA A Level Maths: Mechanics)
Revision Note
Forces in 2D - Vector Notation
How are forces and vectors linked?
- Forces are vectors – they have magnitude and direction.
- The Magnitude of a force is measured in Newtons and the direction of a force is an angle, usually measured in degrees, anti-clockwise from the horizontal
- There are other ways to talk about direction – in particular Cartesian coordinates(x, y) are used to describe the position of a point in two-dimensional space (plane), relative to a fixed origin O.
- You may see questions start with a sentence along the lines of “Relative to the frame of reference Oxy …” – this means the two-dimensional space (plane) and distances within it will be based on horizontal (x) and vertical (y) components relative to the origin O.
What is vector notation?
- Both column vectors and i, j notation can be used for calculating resultant vectors
- In both cases, a vector (force) is written with its two components separated so the magnitude and direction of the vector are not directly known.
What are the notations for magnitude and direction?
- The magnitude of the force F N would be denoted by |F| N or F N – notice the use of bold and italics in particular
- Direction is an angle, usually measured in degrees anti-clockwise from the horizontal, θ° is usually used
How do I find the magnitude and direction of a force from its components?
- For a force F = xi + yj N to find
- its magnitude,|F| N or F N ,use Pythagoras’ theorem
- its direction (as an angle), use a diagram and trigonometry
- If either (or both) components are negative then still use a diagram and trigonometry but
- treat x and y as positive (so |x| and |y| strictly speaking)
- the angle found may need adjusting depending on where, and which way, the direction is being measured from
- e.g. Find the direction at which the force F = (-8i - 6j) N, giving your answer as an angle measured in degrees anti-clockwise from the positive horizontal direction
What does equilibrium with vectors mean?
- In two dimensions a particle is in equilibrium if the resultant force acting on it in both directions is zero
- For column vectors, a resultant force of zero would look like
- For vectors in i-j notation, a resultant force of zero would look like (0i + 0j) N
- Both forms may be written as 0 N (called ‘the zero vector’)
Worked example
(i) Find the resultant force.
(ii) Find the magnitude of the resultant force and its direction as an angle measured
anti-clockwise from the i-direction.
(iii) A third force is applied to the particle such that it is brought into equilibrium.
Find the third force, giving your answer in the form
Examiner Tip
- Ideally you should stick to the i-,j- vector notation used in a question but if you prefer one over the other you can use a mixture within the same question. Just be careful your final answer is in the correct format if requested.
- Draw diagrams – including ‘mini’-diagrams of individual forces/vectors – this can help in understanding a problem and being accurate.
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